Theorem undir 3702

Description: Distributive law for union over intersection. Theorem 29 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)

Assertion

Ref	Expression
undir	|- ( ( A i^i B ) u. C ) = ( ( A u. C ) i^i ( B u. C ) )

Proof of Theorem undir

Step	Hyp	Ref	Expression
1		undi 3700	. 2 |- ( C u. ( A i^i B ) ) = ( ( C u. A ) i^i ( C u. B ) )
2		uncom 3603	. 2 |- ( ( A i^i B ) u. C ) = ( C u. ( A i^i B ) )
3		uncom 3603	. . 3 |- ( A u. C ) = ( C u. A )
4		uncom 3603	. . 3 |- ( B u. C ) = ( C u. B )
5	3, 4	ineq12i 3653	. 2 |- ( ( A u. C ) i^i ( B u. C ) ) = ( ( C u. A ) i^i ( C u. B ) )
6	1, 2, 5	3eqtr4i 2491	1 |- ( ( A i^i B ) u. C ) = ( ( A u. C ) i^i ( B u. C ) )

Syntax hints:   = wceq 1370   u. cun 3429   i^i cin 3430

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-v 3074  df-un 3436  df-in 3438

This theorem is referenced by:  undif1  3857  dfif4  3907  dfif5  3908  bwth  19140